The Uniform Electron Gas Pierre-François Loos* and Peter M

Home , Diffusion Monte Carlo, Jellium, Quantum Monte Carlo, Wigner crystal

Advanced Review The uniform electron gas Pierre-François Loos* and Peter M. W. Gill

The uniform electron gas or UEG (also known as jellium) is one of the most fundamental models in condensed-matter physics and the cornerstone of the most popular approximation—the local-density approximation—within density- functional theory. In this article, we provide a detailed review on the energetics of the UEG at high, intermediate, and low densities, and in one, two, and three dimensions. We also report the best quantum Monte Carlo and symmetry- broken Hartree-Fock calculations available in the literature for the UEG and discuss the phase diagrams of jellium. © 2016 John Wiley & Sons, Ltd

How to cite this article: WIREs Comput Mol Sci 2016, 6:410–429. doi: 10.1002/wcms.1257

INTRODUCTION density correlation functionals (VWN,8 PZ,9 PW92,10 etc.), each of which requires information fi he nal decades of the 20th century witnessed a on the high- and low-density regimes of the spin- Tmajor revolution in solid-state and molecular polarized UEG, and are parametrized using numer- physics, as the introduction of sophisticated 1 ical results from quantum Monte Carlo (QMC) exchange-correlation models propelled density- calculations,11,12 together with analytic perturbative functional theory (DFT) from qualitative to quantita- results. tive usefulness. The apotheosis of this development For this reason, a detailed and accurate under- was probably the award of the 1998 Nobel Prize for 2 3 standing of the properties of the UEG ground state is Chemistry to Walter Kohn and John Pople but its essential to underpin the continued evolution of origins can be traced to the prescient efforts by Tho- DFT. Moreover, meaningful comparisons between mas, Fermi, and Dirac, more than 70 years earlier, to theoretical calculations on the UEG and realistic sys- understand the behavior of ensembles of electrons tems (such as sodium) have also been performed without explicitly constructing their full wave recently (see, e.g., Ref 13). The two-dimensional functions. (2D) version of the UEG has also been the object of In principle, the cornerstone of modern DFT extensive research14,15 because of its intimate connec- – 4 is the Hohenberg Kohn theorem but, in practice, tion to 2D or quasi-2D materials, such as quantum it rests largely on the presumed similarity between dots.16,17 The one-dimensional (1D) UEG has the electronic behavior in a real system and that recently attracted much attention due to its experi- – in the hypothetical three-dimensional (3D) uniform mental realization in carbon nanotubes18 22 organic 5 – electron gas (UEG). This model system was conductors,23 27 transition metal oxides,28 edge applied by Sommerfeld in the early days of quan- 29–31 6 states in quantum Hall liquids, semiconductor tum mechanics to study metals and in 1965, 32–36 fi 37–39 7 heterostructures, con ned atomic gases, Kohn and Sham showed that the knowledge of a and atomic or semiconducting nanowires.40,41 In the analytical parametrization of the UEG correlation present work, we have attempted to collect and col- energy allows one to perform approximate calcula- late the key results on the energetics of the UEG, tions for atoms, molecules, and solids. This information that is widely scattered throughout the spurred the development of a wide variety of spin- physics and chemistry literature. The UEG Paradigm section defines and describes the UEG model in *Correspondence to: [email protected] detail. The High-Density Regime section reports the Research School of Chemistry, Australian National University, known results for the high-density regime, wherein Canberra, Australia the UEG is a Fermi fluid (FF) of delocalized elec- Conflict of interest: The authors have declared no conflicts of inter- trons. The Low-Density Regime section reports est for this article. analogous results for the low-density regime, in

ð which the UEG becomes a Wigner crystal (WC) of ρðÞr vðÞr dr + J½ρ + E =0; ð6Þ relatively localized electrons. The intermediate- b density results from QMC and symmetry-broken Hartree–Fock (SBHF) calculations are gathered in which yields the Intermediate-Density Regime section. Atomic units are used throughout. E½ρ = Ts½ρ + Exc½ρ = T ½ρ + E ½ρ + E ½ρ ð s x ð c ð ð7Þ = ρe ½ρ dr + ρe ½ρ dr + ρe ½ρ dr: UEG PARADIGM t x c

The D-dimensional UEG, or D-jellium, consists of In the following, we will focus on the three reduced interacting electrons in an infinite volume in the pres- (i.e., per electron) energies et, ex, and ec, and we will ence of a uniformly distributed background of posi- discuss these as functions of the Wigner–Seitz radius tive charge. Traditionally, the system is constructed rs defined via by allowing the number n = n" + n# of electrons 8 (where n" and n# are the numbers of spin-up and >4π <> r3, D =3, spin-down electrons, respectively) in a D-dimensional πD=2 s fi 1 D 3 ð Þ cube of volume V to approach in nity with the den- = rs = π 2 8 1 ρ Γ D > rs , D =2, sity ρ = n/V held constant. The spin polarization is 2 +1 > : 2r , defined as s D =1, ρ −ρ − or ζ " # n" n# ; ð Þ = ρ = 1 8 n > 1=3 > 3 , D =3, > 4πρ > where ρ" and ρ# is the density of the spin-up and < 1=2 1 spin-down electrons, respectively, and the ζ = 0 and r = πρ , D =2, ð9Þ s > ζ = 1 cases are called paramagnetic and > 1 > , D =1, ferromagnetic UEGs. :> 2ρ The total ground-state energy of the UEG (including the positive background) is Γ 42 ð where is the Gamma function. It is also conven- ient to introduce the Fermi wave vector E½ρ = Ts½ρ + ρðÞr vðÞr dr + J½ρ + Exc½ρ + Eb; ð2Þ α k = ; ð10Þ F r where Ts is the noninteracting kinetic energy, s ð ρ ðÞr0 where vðÞr = − b dr0 ð3Þ jjr −r0 8 1=3 > 9π , D =3, >p4ffiffiffi 2=D < is the external potential due to the positive back- D−1 D 2, D =2, ρ α =2 D Γ +1 = π ð11Þ ground density b, 2 > > , D =1: ðð : 4 1 ρðÞr ρðÞr0 J½ρ = drdr0 ð4Þ 2 jjr−r0 is the Hartree energy, Exc is the exchange-correlation energy and THE HIGH-DENSITY REGIME In the high-density regime (r 1), also called the ðð 0 s 1 ρ ðÞr ρ ðÞr 0 weakly correlated regime, the kinetic energy of the E = b b drdr ð5Þ b 2 jjr −r0 electrons dominates the potential energy, resulting in a completely delocalized system.5 In this regime, the is the electrostatic self-energy of the positive back- one-electron orbitals are plane waves and the UEG is ground. The neutrality of the system [ρ(r)=ρb(r)] described as a FF. Perturbation theory yields the implies that energy expansion

FFðÞζ ðÞζ ðÞζ FFðÞζ ; ð Þ e rs, = et rs, + ex rs, + ec rs, 12

where the noninteracting kinetic energy et(rs, ζ) and =3

D Eq. (25) Eq. (32b) exchange energy ex(rs, ζ) are the zeroth- and ﬁrst- ) order perturbation energies, respectively, and the cor- ζ ( relation energy eFFðÞr ,ζ encompasses all higher Λ c s ), ζ ( orders. =2 Υ Eq. (39 Eq. (32a) D 11

Noninteracting kinetic energy =1 11—— Eq. (15) Eq. (17c) Eq. (15) — Eq. (17c) —— The noninteracting kinetic energy of D-jellium is the D — ﬁrst term of the high-density energy expansion (12). 2 ⁡

The 3D case has been known since the work of Tho- ln ⁢ 3 3 π 12

43,44 45,46 3 = 4 1 +6 π − mas and Fermi and, for D-jellium, it reads 3 2 2 =

π 24 2 π π 3 4 9 ðÞ 3 3

z = = 4 9

π 1 1 2 3 4 ε ðÞζ 3 π π π 4 t 3 4 4 3 9 9 10 = ðÞζ ; ð Þ 2 ⁡ 1

e r , = 13 − 2 t s 3

2 =3 ln 3 3 = 2 π 2 0.049 917 1 Eq. (44) Ref 58 = = ⁡ r 2 2 − 1 1 6 s 7 4 − 2 2 1 − D 2 ln -jellium at High Density where D (1) 4 λ ðÞ

β 1 2

ε ðÞζ ε Υ ðÞζ ; ð Þ 8 π − t = t t 14a (1), ε π − 10 3 1

π 8 − =2 2 1 4 3 2 2

D ðÞ 2 ⁡

ε ≡ε ðÞζ α ; ð Þ − − =0 = 14b D ln β t t 2ðÞD +2 In 1-jellium, the paramagnetic and ferromagnetic states are degenerate.

and the spin-scaling function is 42 /360

D +2 D +2 2 =1 /24 1 function.

D D ∞ π ðÞζ ðÞ−ζ 2

1+ +1 β π − 00 − 0 00 Υ ðÞζ = : ð15Þ D 0 t 2 2 = 1) States and Spin-Scaling Functions of ⁡

ζ ⁢ The values of εt(ζ) in the paramagnetic and ferro- 3 π 6 12 3 3 3 4

= π − − ðÞ 1 2 2

z is the Dirichlet magnetic limits are given in Table 1 for D =1,2 3 , 24 π π 2 =

π β 2 3 π 4 3 3 9 4 = =

and 3. 1 1

π 2 4 ⁡ π 9 − 3 2 =3 4 ln π π 2

π 4 4 0.071 100 0.010 + 0.008 446 unknown unknown 1 unknown unknown ⁡ 9 9 − 6 3 − 1 − 10 D ln − Exchange Energy 4 1 ðÞ (0)

λ β The exchange energy, which is the second term in − 2

= 0) and Ferromagnetic ( π 8 47,48 π 10 3 1 ζ

(12), can be written (0), − − ﬃﬃ 2

ε π ffiffiffi 2 p 3 =2 2 4 p ðÞ − ε ðÞζ D − β x Paramagnetic State Ferromagnetic State Spin-Scaling Function exðÞrs,ζ = ; ð16Þ rs ) is the Riemann zeta function and n ( z where /360 ln 2 2 =1 /96 1/2 ∞ π 2 0 00 π − − εxðÞζ = εxΥ xðÞζ ; ð17aÞ D 0 cients for the Paramagnetic ( 2D fi ε ≡ε ðÞζ =0 = − α; ð17bÞ x x πðÞ2 −

D 1 cient ﬁ )0 0 ) + 0.008 446 unknown )

) ζ ζ ζ ζ ζ ζ ζ ðÞ ðÞ

ζ ðÞ ðÞ ( (

D +1 D +1 ( ( a b a b 0 1 1 ðÞζ D ðÞ−ζ D t x 0 0 1 1+ +1 Energy Coef ε ε λ ε ε λ λ ε

Υ xðÞζ = : ð17cÞ | 2 is the Euler-Mascheroni constant, γ

s r

s r 2 1 The values of ε (ζ) in the paramagnetic and ferro- ln

x − − s s 0 s s s Term Coef r r ln r r r

magnetic limits are given in Table 1 for D =1,2, and TABLE 1 Note that

3. Note that, due to the particularly strong diver- 3-jellium ε ζ gence of the Coulomb operator, x( ) diverges in 1D. The coefficient λ0(ζ) can been obtained by the Gell- Mann–Brueckner resummation technique,71 which sums the most divergent terms of the series (21) to Hartree–Fock Energy obtain In the high-density limit, one might expect the ð 3 ∞ Hartree–Fock (HF) energy of the UEG to be the sum λ ðÞζ = ½R ðÞu,ζ 2du; ð22Þ 0 π3 0 of the kinetic energy (13) and the exchange energy 32 − ∞ (16), i.e., where eFF ðÞr ,ζ = e ðÞr ,ζ + e ðÞr ,ζ : ð18Þ HF s t s x s u u R0ðÞu,ζ = k#R0 + k"R0 ; ð23aÞ However, although this energy corresponds to a solu- k# k" 5 tion of the HF equation, a stability analysis reveals R0ðÞu =1−u⁢arctanðÞ 1=u ; ð23bÞ that (18) is never the lowest possible HF energy and 49,50 Overhauser showed that it is always possible to and find a symmetry-broken solution of lower energy. We will discuss this further in the Symmetry-broken 1=D k" # =1ðÞ ζ ð24Þ Hartree-Fock section. , In 1D systems, the Coulomb operator is so is the Fermi wave vector of the spin-up or spin-down strongly divergent that a new term appears in the HF electrons. energy expression. Thus, for 1-jellium, Fogler 68 74 The paramagnetic and ferromagnetic limits found51 are given in Table 1, and the spin-scaling function π2 ⁡ ⁢ ðÞπ= − γ h FF 1ln rs 2 ln 2 3+2 1 1 e ðÞrs = − + ; ð19Þ Λ ðÞζ − 3 HF 24r2 2 r 4r 0 = + k#k" k# + k" k# s s s 241ðÞ−ln⁡2 k" k# 42 3 where γ is the Euler–Mascheroni constant. Further- ⁢ln 1 + −k"⁢ln 1 + ð25Þ k# k" more, because the paramagnetic and ferromagnetic states are degenerate in strict 1D systems, we can 76 – was obtained by Wang and Perdew. confine our attention to the latter.52 57 The coefficient ε0(ζ) is often written as the sum

a b ε0ðÞζ = ε ðÞζ + ε ðÞζ ð26Þ Correlation Energy 0 0 The high-density correlation energy expansions of a RPA (random-phase approximation) or ‘ring- ’ εaðÞζ fi diagram term 0 and a rst-order exchange term FF FF e ðÞrs,ζ = erðÞs,ζ −e ðÞrs,ζ ð20Þ εbðÞζ εaðÞζ c HF 0 . The RPA term 0 is not known in closed form but it can be computed numerically with high of the two- and three-dimensional UEGs have been precision.58 Its paramagnetic and ferromagnetic lim- 58–82 well studied. Much less is known about 1-jel- its are given in Table 1 and the spin-scaling function lium.55,83 Using Rayleigh–Schrödinger perturbation theory, the correlation energy appears to possess the Υ aðÞζ εaðÞζ =εaðÞ ð Þ 0 = 0 0 0 27 expansion can be found using Eq. (20) in Ref 58. The first-order X∞ ÂÃ exchange term75 is given in Table 1 and, because it is eFFðÞr ,ζ = λ ðÞζ ln⁡r + ε ðÞζ rj c s j s j s independent of the spin-polarization and the spin- j =0 ð21Þ scaling function = λ0ðÞζ ln⁡rs + ε0ðÞζ + λ1ðÞζ rs⁢ln⁡rs + ε1ðÞζ rs + … Υ bðÞζ εbðÞζ =εbðÞ ð Þ 0 = 0 0 0 =1 28 and the values of these coefficients (when known) are given in Table 1. The methods for their determina- is trivial. tion are outlined in the next three subsections.

ﬁ λ ζ The coef cient 1( ) can be written simi- 3 π2 73 ΛbðÞζ = k2 + k2 larly as 1 π2 −12⁢ln⁡2 6 # " k2 − λ ðÞζ λaðÞζ λbðÞζ ; ð Þ ðÞ− ⁡ − 2 − # k# k" 1 = 1 + 1 29 +1 ln 2 k# k" Li2 2 k# + k" 2 − where k" k" k# 1 4 k# − Li2 + k#⁢ln ð 2 k# + k" k#k" k# + k" α ∞ !#) a 3 a λ ðÞζ = − ℛ ðÞu,ζ du; ð30aÞ k#k" k" 1 π5 1 2 2⁢ 4⁢ ; 8 − ∞ + k#k" ln 2 + k" ln ð k# + k" k# + k" α ∞ λbðÞζ 3 ℛbðÞζ ð Þ ð Þ 1 = 1 u, du 30b 32b 16π4 − ∞ 42 where Li2 is the dilogarithm function. are the RPA and second-order exchange contribu- a b a 10,79 The spin scalings Λ (ζ), Υ ðÞζ , Υ ðÞζ , Λ ðÞζ , tions and α is given in (11). The integrands are 0 0 0 1 ΛbðÞζ and 1 are shown in Figure 1, highlighting the 58 Υ aðÞζ ζ ℛaðÞu,ζ = R ðÞu,ζ 2R ðÞu,ζ ; ð31aÞ Hoffmann minimum in 0 near = 0.9956 and 1 0 1 ΛaðÞζ revealing a similar minimum in 1 near ℛbðÞζ ðÞζ ðÞζ ; ð Þ ζ = 0.9960. It appears that such minima are ubiqui- 1 u, = R0 u, R2 iu, 31b tous in RPA coefﬁcients. −1 u −1 u The data in Table 1 yield the exact values R1ðÞu,ζ = k# R1 + k" R1 ; ð31cÞ k# k" α 7π2 λ ðÞ0 = −12⁢ln⁡2−1 u u 1 π3 R2ðÞiu,ζ = R2 i + R2 i ; ð31dÞ 4 6 k# k" : …; ð Þ π =0009229 33a ðÞ − ; ð Þ R1 u = 2 31e α π2 31+ðÞu2 −4=3 13 1 λ1ðÞ1 =2 −12⁢ln⁡2+ 4π3 12 2 1+3u2 −u 2+3u2 arctan⁡u R ðÞiu =4 : ð31fÞ =0:004792…; ð33bÞ 2 1+u2

73 and it is revealing to compare these with recent Carr and Maradudin gave an estimate of λ1(0) and this was later reﬁned by Perdew et al.10,79 numerical calculations. The estimate λ ≈ 79 However, we have found80 that the integrals in 1(0) 0.0092292 by Sun et al. agrees perfectly λ ≈ Eqs. (30a) and (30b) can be evaluated exactly by with Eq. (33a) but their estimate 1(1) 0.003125 is computer software,84 giving the paramagnetic and strikingly different from Eq. (33b). The error arises ζ ! ferromagnetic values in Table 1 and the spin-scaling from the noncommutivity of the 1 limit and the functions u integration, which is due to the nonuniform con- ℛaðÞζ vergence of 1 u, . 3 π2 1 ΛaðÞζ = + k2 + k2 1 π2 −6 6 4 # " 2 2 3 k# + k" k# − # " − # "⁢ k k 2 2 k k ln 2 k# −k" k" ) 2 2 k# −k" k# −k" k" −k# − Li2 −Li2 ; 2 k# + k" k# + k" ð32aÞ

FIGURE 1 | Spin-scaling functions of 3-jellium as functions of ζ.

Based on the work of Carr and Maradundin,73 and E(x) is the complete elliptic integral of the sec- Endo et al.77 have been able to obtain a numerical value ond kind.42 As in 3-jellium, the constant term ε (ζ) can be ε ðÞ − : ð Þ 0 1 0 = 0 010 34 εaðÞζ decomposed into a direct contribution 0 and a ζ-independent exchange contribution εb for the paramagnetic limit of the term proportional 0 to rs. However, nothing is known about the spin- ε ðÞζ = εaðÞζ + εb: ð41Þ scaling function and the ferromagnetic value for this 0 0 0 coefﬁcient. Calculations by one of the present Following Onsager’s work on the 3D case,75 Isihara authors suggest that the value (34) is probably not and Ioriatti showed63 that accurate,85 mainly due to the large errors in the numerical integrations performed in Ref 73. 8 εb = βðÞ2 − βðÞ4 =+0:114357…; ð42Þ 0 π2

2-jellium where G = β(2) is the Catalan’s constant and β is the 42 Gell-Mann–Brueckner resummation for 2-jellium Dirichlet β function. Recently, we have found 61 εaðÞζ 67 yields closed-form expressions for the direct part 0 . The paramagnetic and ferromagnetic limits are λ0ðÞζ =0; ð35aÞ ð ∞ εaðÞ ⁡ − − : …; ð Þ 1 u u 3 0 0 =ln2 1= 0 306853 43a λ1ðÞζ = − pffiffiffi R + R du; ð35bÞ 12 2π − ∞ k" k# 1 ln⁡2−1 εaðÞ1 = εaðÞ0 = = −0:153426…; ð43bÞ 0 2 0 2 where and the spin-scaling functions are 1 RuðÞ=1− pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: ð36Þ 1+1=u2 −ζ Υ aðÞζ 1 1 ⁢ ⁡ − 0 = + ðÞ⁡ − 2 ln 2 1 2 4ln2 1 sffiffiffiffiffiffiffiffiffiffi! After an unsuccessful attempt by Zia,59 the correct rffiffiffiffiffiffiffiffiffiffi 1+ζ 1+ζ 1−ζ values of the coefficients λ1(0) and λ1(1) were found − + ln 1 + ð44Þ 61 1−ζ 1−ζ 1+ζ by Rajagopal and Kimball to be rffiffiffiffiffiffiffiffiffiffi 1+ζ pffiffiffi 10 −ln 1 + : λ ðÞ0 = − 2 −1 = −0:086314…; ð37Þ 1−ζ 1 3π and Υ bðÞζ = 1. The spin-scaling functions of 2-jellium and74 0 are plotted in Figure 2. To the best of our knowl- pffiffiffi 2 1 10 edge, the term proportional to rs in the high-density λ ðÞ1 = λ ðÞ0 = − −1 = −0:015258…: 1 8 1 4 3π expansion of the correlation energy (21) is unknown for 2-jellium. ð38Þ

Thirty years later, Chesi and Giuliani found66 the spin-scaling function λ1ðÞζ 1 Fk",k# + Fk#,k" Λ1ðÞζ = = k" + k# +3 λ1ðÞ0 8 10−3π ð39Þ where y2 arccos y FxðÞ,y =4ðÞx + y −πx−4xE 1− +2x2 pffiffiffiffiffiffiffiffiffiffiffiffiffix ; x2 x2 −y2 ð40Þ FIGURE 2 | Spin-scaling functions of 2-jellium as functions of ζ.

1-jellium 3-jellium

Again, due to the strong divergence of the Coulomb The leading term of the low-density expansion η0 is the operator in 1D, 1-jellium is peculiar and one has to Madelung constant for the WC.91 In 3D, Coldwell- take special care.57 More details can be found in Ref Horsfall and Maradudin have studied several lattices: 83. The leading term of the high-density correlation simple cubic (sc), face-centered cubic (fcc), and body- energy in 1-jellium has been found to be83 centered cubic (bcc). Carr also mentions92 a calculation for the hexagonal closed pack (hcp) by Kohn and 2 93 π Schechter. The values of η0 for these lattices are ε = − = −0:027416…; ð45Þ 0 360 ηsc − : …; ð Þ 0 = 0 880059 49a and third-order perturbation theory gives73,83 ηhcp − : …; ð Þ 0 = 0 895838 49b ε : : ð Þ 1 =+0008446 46 ηfcc − : …; ð Þ 0 = 0 895877 49c We note that 1-jellium is one of the few systems ηbcc − : …: ð Þ 0 = 0 895930 49d where the rs coefficient of the high-density expansion 77,79 is known accurately. Unlike 2- and 3-jellium, the and reveal that, although all four lattices are energeti- expansion (21) does not contain any logarithm term cally similar, the bcc lattice is the most stable. fi λ λ 92 up to rst order in rs, i.e., 0 = 1 = 0. The high- For the bcc WC, Carr subsequently derived density expansion of the correlation of 1-jellium is the harmonic zero-point energy coefficient

π2 η =1:325; ð50Þ eFFðÞr = − +0:008446r + …: ð47Þ 1 c s 360 s and the ﬁrst anharmonic coefﬁcient94

η − : : ð Þ 2 = 0 365 51 THE LOW-DENSITY REGIME Based on an interpolation, Carr et al.94 estimated the In the low-density (or strongly correlated) regime, the next term of the low-density asymptotic expansion to potential energy dominates over the kinetic energy be η3 ≈ − 0.4. and the electrons localize onto lattice points that Combining Eqs. (49d), (50), and (51) yields the 86,87 minimize their (classical) Coulomb repulsion. low-density energy expansion of the 3D bcc WC These minimum-energy conﬁgurations are called 88 Wigner crystals. In this regime, strong-coupling 0:895930 1:325 0:365 89 WCðÞ− − …: ð Þ methods can be used to show that the WC energy e rs + 3=2 2 + 52 rs r rs has the asymptotic expansion s

X∞ η η η η η 2-jellium eWCðÞr j = 0 + 1 + 2 + 3 + …: ð48Þ s j=2+1 3=2 2 5=2 Following the same procedure as for 3-jellium, Bon- j =0 rs rs rs rs rs sall and Maradundin95 derived the leading term of the low-density energy expansion of the 2D WC for This equation is usually assumed to be strictly inde- the square (□) and triangular (Δ) lattices: pendent of the spin polarization.5,10,79,90 The values of the low-density coefficients for D-jellium are 8 9 < = X0 ÂÃ reported in Table 2 □ 1 2 2 η = − pffiffiffi 2− E− = π ℓ + ℓ 0 π : 1 2 1 2 ; ð53aÞ ℓ1,ℓ2 TABLE 2 | Energy Coefficients of D-jellium at Low Density = −1:100244…; D =3 D =2 D =1 8 9 < = Term Coeff. bcc Lattice Δ Lattice Linear Lattice X0 π Δ 1 2 2 2 η = − pffiffiffi 2− E− = pffiffiffi ℓ −ℓ ℓ + ℓ −1 η − − γ − 0 : 1 2 1 1 2 2 ; r 0 0.895 930 1.106 103 ( ln 2)/2 π 3 s ℓ1,ℓ2 −3=2 η 1.325 0.795 0.359933 rs 1 = −1:106103…; −2 η − 0.365 unknown unknown rs 2 ð53bÞ Note that γ is the Euler-Mascheroni constant.42

416 ©2016JohnWiley&Sons,Ltd Volume6,July/August2016 WIREs Computational Molecular Science The uniform electron gas where techniques97,98 and, in particular, diffusion Monte pffiffiffi pffiffiffi Carlo (DMC) calculations have been valuable in this π ðÞ fi 1 erfc x −x density range. The rst QMC calculations on 2- and E− = ðÞx = pffiffiffi + e ; ð54Þ 11 1 2 x 2 x 3-jellium were reported in 1978 by Ceperley. Although QMC calculations have limitations (finite- 99–103 104–117 erfc is the complementary error function42 and the size effect, fixed-node error, etc), these prime excludes (ℓ1, ℓ2) = (0, 0) from the summation. paved the way for much subsequent research on the 1 This shows that the triangular (hexagonal) lattice is UEG and, indirectly, on the development of DFT. more stable than the square one. For the triangular lattice, Bonsall and Mara- 3-jellium 11 dundin95 also derived the harmonic coefficient Two years after Ceperley’s seminal paper, Ceperley and Alder published QMC results12 that were subse- – η : ; ð Þ quently used by various authors8 10 to construct 1 =0795 55 UEG correlation functionals. In their paper, Ceperley but, to our knowledge, the first anharmonic coeffi- and Alder published released-node DMC results for cient is unknown. This yields the 2D WC energy the paramagnetic and ferromagnetic FF as well as the expression Bose fluid and bcc crystal. Using these data, they proposed the first complete phase diagram of 3-jellium : : and, despite its being based on a Bose bcc crystal, it WCðÞ− 1 106103 0 795 …: ð Þ e rs + 3=2 + 56 is more than qualitatively correct, as we will show rs r s later. In particular, they found that 3-jellium has two phase transitions: a polarization transition (from paramagnetic to ferromagnetic fluid) at rs =75 5 and 1-jellium a ferromagnetic fluid-to-crystal transition at fi fi The rst two coef cients of the low-density energy rs = 100 20. expansion of 1-jellium can be found in Fogler’s In the 1990’s, Ortiz et al. extended Ceperley’s 51 – work. The present authors have also given an alter- study to partially polarized fluid.118 120 They discov- native, simpler derivation using uniformly spaced ered a continuous transition from the paramagnetic 55,96 electrons on a ring. Both constructions lead to to the ferromagnetic state in the range 20 5 ≤ rs ≤ 40 5 and they also predicted a much lower crys- γ −ln⁡2 η = = −0:057966…; ð57aÞ tallization density (rs =65 10) than Ceperley and 0 2 Alder. ð qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 π Using more accurate trial wave function (with η = 2Li ðÞ1 −Li ðÞeiθ −Li ðÞe−iθ dθ fl 121 1 4π 3 3 3 back ow) and twist-averaged boundary condi- 0 tions100 (to minimize finite-size effects), Zong =+0:359933…: ð57bÞ et al.122 re-evaluated the energy of the paramagnetic, ferromagnetic, and partially polarized fluid at rela- 42 where Li3 is the trilogarithm function and the tively low density (40 ≤ rs ≤ 100). They found a energy expansion is second-order transition to a ferromagnetic phase at rs =50 2. According to their results, the ferromag- γ −ln⁡2 0:359933 fl eWCðÞr + + …: ð58Þ netic uid becomes more stable than the paramag- s 3=2 ≈ 2rs rs netic one at rs 80. To complete the picture, Drummond et al.123 reported an exhaustive and meticulous study of the 3D WC over the range 100 ≤ rs ≤ 150. They con- THE INTERMEDIATE-DENSITY cluded that 3-jellium undergoes a transition from a fl REGIME ferromagnetic uid to a bcc WC at rs = 106 1, confirming the early prediction of Ceperley and Quantum Monte Carlo Alder.12 The discrepancy between the crystallization Although it is possible to obtain information on the density found by Ortiz et al.120 and the one deter- high- and low-density limits using perturbation the- mined by Drummond et al.123 is unclear.a The latter ory, this approach struggles in the intermediate- authors have also investigated the possibility of the density regime because of the lack of a suitable refer- existence of an antiferromagnetic WC phase but, ence. As a result, quantum Monte Carlo (QMC) sadly, they concluded that the energy difference

between the ferromagnetic and antiferromagnetic TABLE 3 | Values of the Coefficients a0, a1, and a2 in Eq. (59) and crystals was too small to resolve in their DMC calcu- b0, b1, and b2 in Eq. (60) Used to Parametrize the Energy of 3-jellium lations. More recently, Spink et al.124 have also in the FF and WC Phases reported very accurate DMC energies for the par- Fermi Fluid Wigner Crystal tially polarized fluid phase at moderate density Coefficient Para. Ferro. Coefficient Ferro. (0.2 ≤ rs ≤ 20). The DMC energies of 3-jellium (for the FF and a0 −0.214488 −0.09399 b0 −0.89593

WC phases) have been gathered in Table 4 for vari- a1 1.68634 1.5268 b1 1.3379 ous rs and ζ values. Combining the DMC results of a2 0.490538 0.28882 b2 −0.55270 Zong et al.122 and Drummond et al.,123 we have fl represented the phase diagram of 3-jellium in FF, Fermi uid; WC, Wigner crystal. Figure 3. The correlation energy of the paramagnetic fi and ferromagnetic fluids is fitted using the parametri- Regime section), while b1 and b2 are obtained by t- zation proposed by Ceperley11 ting the DMC results of Ref 123.

FFðÞ paffiffiffiffi0 ; ð Þ 2-jellium ec rs = 59 1+a1 rs + a2rs The first exhaustive study of 2-jellium at the DMC level was published in 1989 by Tanatar and Ceper- 125 where a0, a1, and a2 are fitting parameters. For the ley. In their study, the authors investigate the par- ferromagnetic fluid, we have used the values of a0, amagnetic and ferromagnetic fluid phases, as well as a1, and a2 given in Ref 123. These values have been the ferromagnetic WC with hexagonal symmetry (tri- obtained by fitting the ferromagnetic results of Zong angular lattice). They discovered a Wigner crystalli- 122 fi et al. For the paramagnetic state, we have tted zation at rs =37 5 and they found that, although the paramagnetic results of Ref 122, and found the they are very close in energy, the paramagnetic fluid values given in Table 3. is always more stable than the ferromagnetic one. To parametrize the WC energy data, Drum- Although the Tanatar–Ceperley energies are system- mond et al.123 used another expression proposed by atically too low, as noted by Kwon et al.,126 their Ceperley11 phase diagram is qualitatively correct. A few years later, Rapisarda and Senatore127 b b b revisited the phase diagram of 2-jellium. They found a eWCðÞr = 0 + 1 + 2 : ð60Þ s 3=2 2 region of stability for the ferromagnetic fluid with a rs rs rs polarization transition at rs =20 2 and observed a fl The first coefficient b0 is taken to be equal to the ferromagnetic uid-to-crystal transition at rs =34 low-density limit expansion η0 (see The Low-Density 4. This putative region of stability for the ferromagnetic fluid was also observed by Attaccalite – et al.128 130 who obtained a similar phase diagram with a polarization transition at rs ≈ 26 and a crystallization at rs ≈ 35. An important contribution of Ref 128 was to show that, in contrast to 3-jellium, the partially polarized FF is never a stable phase of 2-jellium. More recently, and in contrast to earlier QMC studies, Drummond and Needs131 obtained statistical errors sufficiently small to resolve the energy difference between the ferromagnetic and paramagnetic fluids. Interestingly, instead of observing a transition from the ferromagnetic fluid to the ferromagnetic crystal, they discovered a transition from the paramagnetic fluid to an antiferromagnetic crystal around rs =31 1. Moreover, they also showed that the ferromagnetic fluid is never more stable than the paramagnetic one, and that it is unlikely that a region of stability exists for a partially spin-polarized fluid. This agrees with the earlier work of Attaccalite FIGURE 3 | Diffusion Monte Carlo phase diagram of 3-jellium. et al.128 However, they did find a transition from the

418 ©2016JohnWiley&Sons,Ltd Volume6,July/August2016 WIREs Computational Molecular Science The uniform electron gas antiferromagnetic to the ferromagnetic WC at have studied 1-jellium qualitatively from the high to 53 rs =38 5. the low-density regimes. Lee and Drummond have Some authors have investigated the possibility published accurate DMC data for the range 1 ≤ rs ≤ of the existence of a ‘hybrid phase’ in the vicinity of 20. The present authors have published DMC data the transition density from ferromagnetic fluid to fer- at higher and lower densities in order to parametrize – romagnetic WC.131 135 According to Falakshahi and a generalized version of the LDA.55,56,96 The DMC Waintal,133,134 the hybrid phase has the same sym- data for 1-jellium are reported in Table 6. metry as the WC but has partially delocalized orbi- Using the ‘robust’ interpolation proposed by tals. However, its existence is still under debate.131 Cioslowski136 and the high- and low-density expan- The DMC energies of 2-jellium (for the fluid sions (47) and (58), the correlation energy of 1- and crystal phases) have been gathered in Table 4 for jellium calculated with the HF energy given by various rs. Based on the data of Ref 131, we have con- (19) can be approximated by structed the phase diagram of 2-jellium in Figure 4. fl fi X3 The uid energy data are tted using the parametriza- − 127 LDAðÞ 2 jðÞ− 3 j; ð Þ tion proposed by Rapisarda and Senatore: ec rs = t cjt 1 t 64 j =0 pffiffiffiffi r + a FFðÞ ⁢ ps ffiffiffiffi 1 ec rs = a0 1+Ars B ln with pffiffiffiffirs C r +2a r + a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + ln s 2 s 3 1+4krs −1 20 rs 139 ð61Þ t = ; ð65Þ pffiffiffiffi => 2krs B r + a πC7 @ qffiffiffiffiffiffiffiffiffiffiffiffiffis 2 − A5 ; + D arctan > − 2 2 ; and a3 a2 c = kη , c =4kη + k3=2η ; ð66aÞ where 0 0 1 0 1 c2 =5ε0 + ε1=k, c3 = ε1; ð66bÞ 2ðÞa +2a 1 1 A = 1 2 , B = − ; ð62aÞ − − 2 where k = 0.414254 is a scaling factor which is 2a1a2 a3 a1 a1 a1 +2a2 determined by a least-squares fit of the DMC data a 2a 1 F−a C C = 1 − 2 + , D = qffiffiffiffiffiffiffiffiffiffiffiffiffi2 ; ð62bÞ given in Refs 53 and 55. a3 a3 a1 +2a2 − 2 The results using the LDA correlation func- a3 a2 tional (64) are compared to the DMC calculations of ðÞ− 1 − 2a2 : ð Þ Refs 53 and 55. The results are gathered in Table 7 F =1+ 2a2 a1 62c ≤ ≤ a1 +2a2 a3 and depicted in Figure 5. For 0.2 rs 100, the LDA and DMC correlation energies agree to within 0.1 millihartree, which is remarkable given the sim- To parametrize the WC energies, Drummond plicity of the functional. and Needs131 used the expression proposed by Ceperley11 Symmetry-broken Hartree-Fock 49,50 WCðÞ b0 b1 b2 b3 b4 : ð Þ In the early 1960s, Overhauser showed that the e rs = + 3=2 + 2 + 5=2 + 3 63 rs rs rs rs rs HF energy (18) for the paramagnetic FF can always be improved by following spin- and charge-density 5 The first two coefficients b0 and b1 are taken to be instabilities to locate a SBHF solution. Recently, a ‘ ’ equal to the low-density limit expansion η0 and η1 (see computational proof has been given by Zhang and The Low-Density Regime section), and the others are Ceperley137 who performed unrestricted HF (UHF) found by fitting to their DMC results. The values of calculations on the paramagnetic state of finite-size the fitting coefficients for 2-jellium are given in Table 5. 3D UEGs and discovered broken spin-symmetry solutions, even for high densities. In 2D, this has been proven rigorously for the ferromagnetic state by 1-jellium Bernu et al.138 The first phase diagrams based on Not surprisingly, there have been only a few QMC UHF calculations for 2- and 3-jellium were per- studies on 1-jellium. Astrakharchik and Girardeau52 formed by Trail et al.139 who found lower energies

TABLE 4 | DMC Energy of 3-jellium at Various rs for the FF and WC Phases Partially Spin-Polarized Fluid rs Para. Fluid Ferro. Fluid Ferro. Crystal ζ =0 ζ = 0.185 ζ = 0.333 ζ = 0.519 ζ = 0.667 ζ = 0.852 ζ =1 ζ =1 0.5 3.430 11(4) — 3.692 87(6) — 4.441 64(6) — 5.824 98(2) — 1 0.587 80(1) — 0.649 19(2) — 0.823 94(4) — 1.146 34(2) — 2 0.002 380(5) — 0.016 027(6) — 0.054 75(2) — 0.126 29(3) — 3 −0.067 075(4) — −0.061 604(5) — −0.046 08(2) — −0.017 278(4) — 06Jh ie os t oue6 uyAgs 2016 July/August 6, Volume Ltd Sons, & Wiley John 2016 © 5 −0.075 881(1) — −0.074 208(4) — −0.069 548(4) — −0.060 717(5) — 10 −0.053 511 6(5) — −0.053 214(2) — −0.052 375(2) — −0.050 733 7(5) — 20 −0.031 768 6(5) — −0.031 715 6(7) — −0.031 594 0(7) — −0.031 316 0(4) — 40 −0.017 618 7(3) — −0.017 6165(3) — −0.017 602 7(3) — −0.017 567 4(4) — 50 −0.014 449 5(3) −0.014 449 5(3) −0.014 449 8(3) −0.014 447 3(4) −0.014 444 2(3) −0.014 437 7(4) −0.014 424 9(4) — 60 −0.012 260 1(2) −0.012 259 3(3) −0.012 260 2(2) −0.012 259 8(3) −0.012 258 7(2) −0.012 255 9(2) −0.012 250 8(2) — 70 −0.010 657 2(2) −0.010 656 9(2) −0.010 658 1(2) −0.010 658 6(3) −0.010 658 0(2) −0.010 656 7(2) −0.010 653 3(2) — 75 −0.010 005 7(2) −0.010 006 0(2) −0.010 006 9(2) — −0.010 007 2(2) — −0.010 004 4(2) — 85 −0.008 920 1(2) — −0.008 9208(2) — −0.008 921 5(2) — −0.008 920 6(2) — 100 −0.007 676 8(2) — −0.007 677(2) — −0.007 678 2(1) — −0.007 678 8(1) −0.007 676 5(4) 110 ———————−0.007 031 2(5) 125 ———————−0.006 245 8(4) 150 ———————−0.005 269 0(3)

For the FF, the data from rs = 0.5 to 20 are taken from Ref 124, and the data from rs = 40 to 100 are taken from Ref 122. The data for the ferromagnetic WC are taken from Ref 123. The statistical error is reported in parenthesis. ﬂ

DMC, Diffusion Monte Carlo; FF, Fermi uid; WC, Wigner crystal. wires.wiley.com/compmolsci WIREs Computational Molecular Science The uniform electron gas

for a crystal for rs > 1.44 in 2D and rs > 4.5 in 3D. Curiously, as we will show below, the SBHF phase diagram is far richer than the near-exact DMC one presented in the Quantum Monte Carlo section. Before going further, it is interesting to investigate the HF expression of the FF given by (18), and study the phase diagram based on this simple expres- sion5 (see Figure 6 for the example of 3-jellium). It is B easy to show that, for 0 < rs < rs , the paramagnetic ﬂuid is predicted to be lower in energy than the ferromagnetic ﬂuid where 8 = <2:011, D =2, 22 D −1 ε rB = − t = ð67Þ s 1=D ε : 2 −1 x 5:450, D =3,

and ε and ε are given by Eqs. (14b) and (17b), FIGURE 4 t x | Diffusion Monte Carlo phase diagram of 2-jellium. respectively. This sudden paramagnetic-to-

TABLE 5 | Values of the Coefficients a0, a1, a2, and a3 in Eq. (61) and b0, b1, b2 and b3 and b4 in Eq. (63) Used to Parameterize the Energy of 2-jellium in the FF and WC Phases Fermi Fluid Wigner Crystal Value Value Coefficient Para. Fluid Ferro. Fluid Coefficient Ferro. Crystal Antif. Crystal

a0 −0.186 305 2 −0.290 910 2 b0 −1.106 103 −1.106 103

a1 6.821 839 −0.624 383 6 b1 0.814 0.814

a2 0.155 226 1.656 628 b2 0.113 743 0.266 297 7

a3 3.423 013 3.791 685 b3 −1.184 994 −2.632 86

b4 3.097 610 6.246 358 FF, Fermi ﬂuid; WC, Wigner crystal.

TABLE 6 | DMC Energy of 2-jellium at Various rs for the FF and WC Phases

rs Para. Fluid Ferro. Fluid Antif. Crystal Ferro. Crystal ζ =0 ζ =1 ζ =0 ζ =1 1 − 0.209 8(3) ——— 5 − 0.149 5(1) − 0.143 3(1) —— 10 − 0.085 36(2) − 0.084 48(4) —— 15 ———− 0.059 665(1) 20 − 0.046 305(4) − 0.046 213(3) − 0.046 229(2) − 0.046 195(2) 25 − 0.037 774(2) − 0.037 740(2) − 0.037 751(3) − 0.037 731(2) 30 − 0.031 926(1) − 0.031 913(1) − 0.031 922(2) − 0.031 917(2) 35 − 0.027 665(1) − 0.027 657(1) − 0.027 672(1) − 0.027 669(1) 40 − 0.024 416(1) − 0.024 416(1) − 0.024 431(2) − 0.024 432(1) 45 ——− 0.021 875(2) − 0.021 881(1) 50 ——− 0.019 814(2) − 0.019 817(2)

Data from rs = 1 to 10 are taken from Ref 126 for the paramagnetic fluid. Data from rs = 5 to 10 are taken from Ref 127 for the ferromagnetic fluid. Data from rs = 15 to 50 are taken from Ref 131. The statistical error is reported in parenthesis. DMC, diffusion Monte Carlo; FF, Fermi fluid; WC, Wigner crystal.

TABLE 7 | DMC Energy and Reduced Energy Given by Eq. (66a) + B The fact that rs > rs implies that this state is for 1-jellium at Various rs locally stable with respect to partial spin polarization FF LDA and will not undergo a continuous phase transition rs DMC Energy e + e HF c to the ferromagnetic state, in contrast to the predic- 0.2 13.100 54(2) 13.100 53 tions of DMC calculations on 3-jellium, as discussed 0.5 1.842 923(2) 1.842 850 in the Quantum Monte Carlo section. 1 0.154 188 6(2) 0.154 101 4 B For rs > rs , the ferromagnetic state is lower in 2 −0.206 200 84(7) −0.206 219 38 energy than the paramagnetic state. However, a simi- 5 −0.203 932 35(2) −0.203 843 14 lar stability analysis yields 10 −0.142 869 097(9) −0.142 781 622 15 −0.110 466 761(4) −0.110 400 702 eFF ðÞr , ζ = eFF ðÞr ,1 HF s HF s 20 −0.090 777 768(2) −0.090 727 757 ε ε −ðÞ−ζ D +2 t D +1 x 50 −0.046 144(1) −0.046 128 1 D−1 2 + D−1 ð70Þ D r D r 2 D s 2 D s 100 −0.026 699(1) −0.026 694 D +1 + O ðÞ1−ζ D ; The DMC data from rs = 1 to 20 are taken from Ref 53. The rest is taken from Refs. 55,83,96. The statistical error is reported in parenthesis. DMC, diffusion Monte Carlo. which shows that the ferromagnetic state is never a − ferromagnetic transition is sometimes called a Bloch stationary minimum. In fact, for rs < rs , where 140 transition. Expanding the HF expression of the 8 paramagnetic state around ζ = 0 yields <1:571, D =2, r + r− = s = ð71Þ s D−1 : 2 D 3:798, D =3, FF ðÞζ FF ðÞ e rs, = e rs,0 HF HF D +2εt D +1εx ð68Þ + ζ2 + + O ζ4 ; the ferromagnetic state is locally unstable and can D2 r2 2D2 r s s undergo a continuous depolarization toward the paramagnetic state. Taken together, these predictions and reveals that this state is locally stable with respect to partial spin polarization until 8 : ðÞε <2 221, D =2, + − 2 D +2 t ð Þ rs = ε = : 69 D +1 x 6:029, D =3:

FIGURE 6 | FFðÞζ eHF rs, as a function of rs for the paramagnetic and fl B ferromagnetic uid phases of 3-jellium (see Eq. (18)). For rs > rs , the ferromagnetic fluid becomes lower in energy than the paramagnetic FIGURE 5 | LDAðÞ fl − fl ec rs of 1-jellium given by Eq. (66a) as a function uid (Bloch transition). For rs < rs , the ferromagnetic uid becomes + of rs (solid line). Diffusion Monte Carlo results from Table 7 are shown locally unstable toward depolarization, while for rs > rs , the by black dots. The small-rs expansion of Eq. (49a) (dashed line) and paramagnetic fluid becomes locally unstable toward polarization. The large-rs approximation of Eq. (60) (dotted line) are also shown. ‘hysteresis loop’ is indicated in red.

imply the ‘hysteresis loop’ shown in Figure 6 for 3-jellium.

3-jellium Baguet et al.141,142 have obtained what is thought to be the complete phase diagram of 3-jellium at the HF level. The SBHF phase diagram of 3-jellium is represented in Figure 7 using the data reported in Refs 141,142 (see Table 8). In addition to the usual FF and WC phases, they have also considered incommensurate crystals (ICs) with sc, fcc, bcc, and hcp unit cells. In an IC, the number of maxima of the charge density is higher than the number of electrons, having thus metallic character. As one can see in FIGURE 7 | SBHF phase diagram of 3-jellium constructed with the Figure 7, the phase diagram is complicated and, data of Refs 141 and 142, (see Table 8). unfortunately, ﬁnite-size effects prevent a precise

TABLE 8 | SBHF Energy (in Millihartree) of 3-jellium for Various rs Values

rs Energy Lattice Phase Polarization rs Energy Lattice Phase Polarization 3.0 −29.954 bcc IC Para. 7.6 −47.804 sc WC Para. 3.1 −32.826 bcc IC Para. 7.8 −47.403 sc WC Para. 3.2 −35.289 bcc IC Para. 8.0 −46.992 sc WC Para. 3.3 −37.399 bcc IC Para. 8.2 −46.576 sc WC Para. 3.4 −39.287 hcp WC Para. 8.4 −46.155 sc WC Para. 3.5 −40.923 hcp WC Para. 8.6 −45.731 sc WC Para. 3.6 −42.437 hcp WC Para. 8.8 −45.307 sc WC Para. 3.7 −43.727 fcc WC Para. 9.0 −44.883 sc WC Para. 3.8 −44.899 fcc WC Para. 9.2 −44.461 sc WC Para. 4.0 −46.775 fcc WC Para. 9.4 −44.050 hcp WC Ferro. 4.2 −48.157 fcc WC Para. 9.6 −43.647 hcp WC Ferro. 4.4 −49.151 fcc WC Para. 9.8 −43.245 hcp WC Ferro. 4.6 −49.841 fcc WC Para. 10.0 −42.844 hcp WC Ferro. 4.8 −50.292 fcc WC Para. 10.2 −42.444 hcp WC Ferro. 5.0 −50.554 fcc WC Para. 10.4 −42.047 fcc WC Ferro. 5.2 −50.665 fcc WC Para. 10.7 −41.461 fcc WC Ferro. 5.4 −50.656 fcc WC Para. 11.0 −40.883 fcc WC Ferro. 5.6 −50.551 fcc WC Para. 11.5 −39.936 fcc WC Ferro. 5.8 −50.368 fcc WC Para. 12.0 −39.015 fcc WC Ferro. 6.0 −50.192 sc WC Para. 12.5 −38.126 fcc WC Ferro. 6.2 −50.031 sc WC Para. 13.0 −37.267 fcc WC Ferro. 6.4 −49.813 sc WC Para. 13.5 −36.441 bcc WC Ferro. 6.6 −49.550 sc WC Para. 14.0 −35.645 bcc WC Ferro. 6.8 −49.249 sc WC Para. 14.5 −34.880 bcc WC Ferro. 7.0 −48.919 sc WC Para. 15.0 −34.142 bcc WC Ferro. 7.2 −48.566 sc WC Para. 15.5 −33.432 bcc WC Ferro. 7.4 −48.193 sc WC Para. 16.0 −32.748 bcc WC Ferro.

− The energy data are taken from the supplementary materials of Ref 141. The precision of the calculations is of the order 5 × 10 3 millihartree. SBHF, symmetry-broken Hartree–Fock; bcc, body-centered cubic; fcc, face-centered cubic; hcp, hexagonal closed pack; IC, incommensurate crystal; sc, simple cubic; WC, Wigner crystal.

determination of the ground state for rs < 3. How- (a) ever, extending the analysis of Ref 143, one can prove that the incommensurate phases are always energetically lower than the FF in the high-density limit. This particular point has been recently discussed in Ref 144. For 3 < rs < 3.4, the incommensurate metallic phase with a bcc lattice is found to be the lowest- energy state. For rs > 3.4, the 3-jellium ground state is a paramagnetic WC with hcp (3.4 < rs < 3.7), fcc (3.7 < rs < 5.9), and sc (5.9 < rs < 9.3) lattices. From any value of rs greater than 9.3, the ground state is a ferromagnetic WC with hcp (9.3 < rs < 10.3), fcc (10.3 < rs < 13), and ﬁnally bcc (rs > 13) lattices. It is interesting to note that, compared to the DMC (b) results from the Quantum Monte Carlo section, at the HF level, the Wigner crystallization happens at much higher densities, revealing a key deﬁciency of the HF theory.

2-jellium In 2D, Bernu et al.145 have obtained the SBHF phase diagram by considering the FF, the WC, and the IC with square or triangular lattices. The phase diagram is shown in Figure 8. They have shown that the incommensurate phase is always favored compared to the FF, independently of the imposed polarization and crystal symmetry, in agreement with the early prediction of Overhauser about the instability of the FIGURE 8 | Left: Symmetry-broken Hartree–Fock (SBHF) phase FF phase.49,50 The paramagnetic incommensurate diagram of 2-jellium constructed with the data of Ref 145. Right: hexagonal crystal is the true HF ground state at high SBHF in the high-density region (0 < rs < 1.3). densities (rs < 1.22). For rs > 1.22, the paramagnetic incommensurate hexagonal crystal becomes a com- mensurate WC of hexagonal symmetry, and at rs ≈ – 1.6, a structural transition from the paramagnetic Monte Carlo calculations.146 148 The finite- hexagonal WC to the ferromagnetic square WC temperature UEG is of key relevance for many appli- occurs, followed by a transition from the paramag- cations in dense plasmas, warm dense matter, and netic square WC to the ferromagnetic triangular WC finite-temperature DFT.149,150 at rs ≈ 2.6. Interestingly, as at the DMC level (see Quantum Monte Carlo section), they do not find a stable partially polarized state. CONCLUSION 1-jellium Mark Twain once wrote, ‘There is something fasci- To the best of our knowledge, the SBHF phase dia- nating about science. One gets such wholesale returns fl gram of 1-jellium is unknown, but it would probably of conjecture out of such a tri ing investment of ’ be very instructive. fact. How true this is of the UEG! We have no simpler paradigm for the study of large numbers of interacting electrons and yet, out of that simplicity, Finite-Temperature Calculations behavior of such complexity emerges that the UEG All the results reported in the present review con- has become one of the most powerful pathways for cerned the UEG at zero temperature. Recently, par- rationalizing and predicting the properties of atoms, ticular efforts have been devoted to obtain the molecules, and condensed-phase systems. The beauty properties of the finite-temperature UEG in the of this unexpected ex nihilo complexity has lured warm-dense regime using restricted path-integral many brilliant minds over the years and yet it is a

424 ©2016JohnWiley&Sons,Ltd Volume6,July/August2016 WIREs Computational Molecular Science The uniform electron gas siren song for, 90 years after the publication of will improve our understanding of, and our ability to Schrödinger’s equation, a complete understanding of model, phase transitions in large quantum mechani- the UEG (even in the nonrelativistic limit) continues cal systems. to elude quantum scientists. Many regard a full treatment of the UEG as In this review, we have focused on the energy one of the major unsolved problems in quantum sci- of the UEG, rather than on its many other interesting ence. We hope that, by providing a snapshot of the properties. We have done so partly for the sake of state of the art in 2016, we will inspire the next gen- brevity and partly because most properties can be eration to roll up their sleeves and confront this fasci- cast as derivatives of the energy with respect to one nating challenge. or more external parameters. Such properties are attracting increasing attention in their own right and we look forward to comprehensive reviews on these NOTE in the years ahead. However, we also foresee contin- a The difference between the crystallization densities ued developments in the accurate calculations of the reported in Refs 120 and 12 is less than two error bars, energies themselves. These will play a critical role in whereas the crystallization density difference between Refs the ongoing evolution of QMC methodology and 120 and 123 is of greater signiﬁcance.

ACKNOWLEDGMENTS The authors would like to thank Neil Drummond, Mike Towler, and John Trail for useful discussions, and Bernard Bernu and Lucas Baguet for providing the data for the phase diagram of 2-jellium. P.F.L. thanks the Australian Research Council for a Discovery Early Career Researcher Award (Grant No. DE130101441) and a Discovery Project grant (DP140104071). P.M.W.G. thanks the Australian Research Council for funding (Grants No. DP120104740 and DP140104071).

REFERENCES 1. Parr RG, Yang W. Density-Functional Theory of 10. Perdew JP, Wang Y. Accurate and simple analytic Atoms and Molecules. Oxford: Clarendon Press; 1989. representation of the electron-gas correlation energy. – 2. Kohn W. Nobel lecture: electronic structure of Phys Rev B 1992, 45:13244 13249. matter—wave functions and density functionals. Rev 11. Ceperley DM. Ground state of the fermion one- Mod Phys 1999, 71:1253–1266. component plasma: a Monte Carlo study in two and 3. Pople JA. Nobel lecture: quantum chemical models. three dimensions. Phys Rev B 1978, 18:3126–3138. – Rev Mod Phys 1999, 71:1267 1274. 12. Ceperley DM, Alder BJ. Ground state of the electron 4. Hohenberg P, Kohn W. Inhomogeneous electron gas. gas by a stochastic method. Phys Rev Lett 1980, Phys Rev 1964, 136:B864–B871. 45:566–569. 5. Giuliani GF, Vignale G. Quantum Theory of the Elec- 13. Huotari S, Soininen JA, Pylkkäanen T, tron Liquid. Cambridge: Cambridge University Hämäläinen K, Issolah A, Titov A, McMinis J, Kim J, Press; 2005. Esler K, Ceperley DM, et al. Momentum distribution 6. Sommerfeld A. Zur elektronentheorie der metalle auf and renormalization factor in sodium and the electron grund der fermischen statistik. Zeits fur Physik 1928, gas. Phys Rev Lett 2010, 105:086403. – 47:1 32. 14. Ando T, Fowler AB, Stern F. Electronic properties of 7. Kohn W, Sham LJ. Self-consistent equations including two-dimensional systems. Rev Mod Phys 1982, exchange and correlation effects. Phys Rev 1965, 54:437–672. 140:A1133–A1138. 15. Abrahams E, Kravchenko SV, Sarachik MP. Metallic 8. Vosko SH, Wilk L, Nusair M. Accurate spin-depend- behavior and related phenomena in two dimensions. ent electron liquid correlation energies for local spin Rev Mod Phys 2001, 73:251–266. density calculations: a critical analysis. Can J Phys 1980, 58:1200. 16. Alhassid Y. The statistical theory of quantum dots. Rev Mod Phys 2000, 72:895–968. 9. Perdew JP, McMullen ER, Zunger A. Density- functional theory of the correlation energy in atoms 17. Reimann SM, Manninen M. Electronic structure of – and ions: a simple analytic model and a challenge. quantum dots. Rev Mod Phys 2002, 74:1283 1342. Phys Rev A 1981, 23:2785–2789.

18. Saito R, Dresselhauss G, Dresselhaus MS. Properties 32. Gonï AR, Pinczuk A, Weiner JS, Calleja JM, of Carbon Nanotubes. London: Imperial College Dennis BS, Pfeiffer LN, West KW. One-dimensional Press; 1998. plasmon dispersion and dispersionless intersubband 19. Egger R, Gogolin AO. Correlated transport and non- excitations in gaas quantum wires. Phys Rev Lett – Fermi-liquid behavior in single-wall carbon nano- 1991, 67:3298 3301. tubes. Eur Phys J B 1998, 3:281–300. 33. Auslaender OM, Yacoby A, dePicciotto R, Baldwin KW, Pfeiffer LN, West KW. Experimental 20. Bockrath M, Cobden DH, Lu J, Rinzler AG, evidence for resonant tunneling in a Luttinger liquid. Smalley RE, Balents L, McEuen PL. Luttinger-liquid Phys Rev Lett 2000, 84:1764–1767. behaviour in carbon nanotubes. Nature 1999, 397:598–601. 34. Zaitsev-Zotov SV, Kumzerov YA, Firsov YA, Monceau P. Luttinger-liquid-like transport in long 21. Ishii H, Kataura H, Shiozawa H, Yoshioka H, InSb nanowires. J Phys Condens Matter 2000, 12: Otsubo H, Takayama Y, Miyahara T, Suzuki S, L303–L309. Achiba Y, Nakatake M, et al. Direct observation of Tomonaga–Luttinger-liquid state in carbon nanotubes 35. Liu F, Bao M, Wang KL, Li C, Lei B, Zhou C. One- at low temperatures. Nature 2003, 426:540–544. dimensional transport of In2O3 nanowires. Appl Phys Lett 2005, 86:213101. 22. Shiraishi M, Ata M. Tomonaga–Luttinger-liquid 36. Steinberg H, Auslaender OM, Yacoby A, Qian J, behavior in single-walled carbon nanotube networks. Fiete GA, Tserkovnyak Y, Halperin BI, Baldwin KW, Solid State Commun 2003, 127:215–218. Pfeiffer LN, West KW. Localization transition in a 23. Schwartz A, Dressel M, Grüuner G, Vescoli V, ballistic quantum wire. Phys Rev B 2006, 73:113307. Degiorgi L, Giamarchi T. On-chain electrodynamics 37. Monien H, Linn M, Elstner N. Trapped one- of metallic (TMTSF) X salts: observation of 2 dimensional Bose gas as a Luttinger liquid. Phys Rev Tomonaga–Luttinger liquid response. Phys Rev B A 1998, 58:R3395–R3398. 1998, 58:1261–1271. 38. Recati A, Fedichev PO, Zwerger W, Zoller P. Fermi 24. Vescoli V, Zwick F, Henderson W, Degiorgi L, one-dimensional quantum gas: Luttinger liquid Grioni M, Gruner G, Montgomery LK. Optical and approach and spin–charge separation. J Opt B Quan- – photoemission evidence for a Tomonaga Luttinger tum Semiclassical Opt 2003, 5:S55–S64. liquid in the Bechgaard salts. Eur Phys J B 2000, 39. Moritz H, Stoferle T, Guenter K, Kohl M, 13:503–511. Esslinger T. Confinement induced molecules in a 1D 25. Lorenz T, Hofmann M, Grüninger M, Freimuth A, Fermi gas. Phys Rev Lett 2005, 94:210401. Uhrig GS, Dumm M, Dressel M. Evidence for spin– 40. Schäafer J, Blumenstein C, Meyer S, Wisniewski M, charge separation in quasi-one-dimensional organic Claessen R. New model system for a one-dimensional conductors. Nature 2002, 418:614–617. electron liquid: self-organized atomic gold chains on 26. Dressel M, Petukhov K, Salameh B, Zornoza P, Ge(001). Phys Rev Lett 2008, 101:236802. Giamarchi T. Scaling behavior of the longitudinal 41. Huang Y, Duan X, Cui Y, Lauhon LJ, Kim K-H, and transverse transport in quasi-one-dimensional Lieber CM. Logic gates and computation from organic conductors. Phys Rev B 2005, 71:075104. assembled nanowire building blocks. Science 2001, 27. Ito T, Chainani A, Haruna T, Kanai K, Yokoya T, 294:1313–1317. Shin S, Kato R. Temperature-dependent Luttinger 42. Olver FWJ, Lozier DW, Boisvert RF, Clark CW, eds. surfaces. Phys Rev Lett 2005, 95:246402. NIST Handbook of Mathematical Functions. 28. Hu Z, Knupfer M, Kielwein M, RoÏer UK, New York: Cambridge University Press; 2010. Golden MS, Fink J, de Groot FMF, Ito T, Oka K, 43. Thomas LH. The calculation of atomic fields. Proc Kaindl G. The electronic structure of the doped one- Camb Philos Soc 1927, 23:542–548. dimensional transition metal oxide Y2-xCaxBaNiO5 44. Fermi E. Un metodo statistico per la determinazione studied using X-ray absorption. Eur Phys J B 2002, di alcune priopriet a dell’atomo. Rend Accad Naz – 26:449 453. Lincei 1927, 6:602–607. 29. Milliken FP, Umbach CP, Webb RA. Indications of a 45. Glasser ML, Boersma J. Exchange energy of an elec- Luttinger liquid in the fractional quantum Hall tron gas of arbitrary dimensionality. SIAM J Appl – regime. Solid State Commun 1996, 97:309 313. Math 1983, 43:535–545. 30. Mandal SS, Jain JK. How universal is the fractional- 46. Iwamoto N. Sum rules and static local-field correc- quantum-Hall edge Luttinger liquid? Solid State Com- tions of electron liquids in two and three dimensions. mun 2001, 118:503–507. Phys Rev A 1984, 30:3289–3304. 31. Chang AM. Chiral Luttinger liquids at the fractional 47. Dirac PAM. Note on exchange phenomena in the quantum Hall edge. Rev Mod Phys 2003, Thomas–Fermi atom. Proc Camb Philos Soc 1930, 75:1449–1505. 26:376–385.

48. Friesecke G. Pair correlations and exchange phenom- 67. Loos PF, Gill PMW. Exact energy of the spin- ena in the free electron gas. Commun Math Phys polarized two-dimensional electron gas at high den- 1997, 184:143–171. sity. Phys Rev B 2011, 83:233102. 49. Overhauser AW. New mechanism of antiferromag- 68. Macke W. Uber die wechselwirkungen im Fermi-gas. netism. Phys Rev Lett 1959, 3:414–416. Z Naturforsch A 1950, 5a:192–208. 50. Overhauser AW. Spin density waves in an electron 69. Bohm D, Pines D. A collective description of electron gas. Phys Rev 1962, 128:1437–1452. interactions: III. Coulomb interactions in a degenerate electron gas. Phys Rev 1953, 92:609–625. 51. Fogler MM. Ground-state energy of the electron liq- 70. Pines D. A collective description of electron interac- uid in ultrathin wires. Phys Rev Lett 2005, tions: IV. Electron interaction in metals. Phys Rev 94:056405. 1953, 92:626–636. 52. Astrakharchik GE, Girardeau MD. Exact ground- 71. Gell-Mann M, Brueckner KA. Correlation energy of state properties of a one-dimensional Coulomb gas. an electron gas at high density. Phys Rev 1957, Phys Rev B 2011, 83:153303. 106:364–368. 53. Lee RM, Drummond ND. Ground-state properties of 72. DuBois DF. Electron interactions. Part I. Field theory the one-dimensional electron liquid. Phys Rev B of a degenerate electron gas. Ann Phys 1959, 2011, 83:245114. 7:174–237. 54. Loos PF, Gill PMW. Exact wave functions of two- 73. Carr WJ, Maradudin AA. Ground-state energy of a electron quantum rings. Phys Rev Lett 2012, high-density electron gas. Phys Rev 1964, 133: 108:083002. A371–A374. 55. Loos PF, Gill PMW. Uniform electron gases. 74. Misawa S. Ferromagnetism of an electron gas. Phys I. Electrons on a ring. J Chem Phys 2013, Rev 1965, 140:A1645–A1648. 138:164124. 75. Onsager L, Mittag L, Stephen MJ. Integrals in the 56. Loos PF, Ball CJ, Gill PMW. Uniform electron gases. theory of electron correlations. Ann Phys 1966, – II. The generalized local density approximation in 18:71 77. one dimension. J Chem Phys 2014, 140:18A524. 76. Wang Y, Perdew JP. Spin scaling of the electron-gas 57. Loos PF, Ball CJ, Gill PMW. Chemistry in one dimen- correlation energy in the high-density limit. Phys Rev – sion. Phys Chem Chem Phys 2015, 17:3196–3206. B 1991, 43:8911 8916. 77. Endo T, Horiuchi M, Takada Y, Yasuhara H. High- 58. Hoffman GG. Correlation energy of a spin-polarized density expansion of correlation energy and its electron gas at high density. Phys Rev B 1992, extrapolation to the metallic density region. Phys Rev 45:8730–8733. B 1999, 59:7367–7372. 59. Zia RKP. Electron correlation in a high density gas 78. Ziesche P, Cioslowski J. The three-dimensional elec- fi fi con ned to very thin lms. J Phys C 1973, tron gas at the weak-correlation limit: how peculiari- – 6:3121 3129. ties of the momentum distribution and the static 60. Isihara A, Toyoda T. Correlation energy of 2-D elec- structure factor give rise to logarithmic non- trons. Ann Phys 1977, 106:394–406. analyticities in the kinetic and potential correlation energies. Physica A 2005, 356:598–608. 61. Rajagopal AK, Kimball JC. Correlations in a two- dimensional electron system. Phys Rev B 1977, 79. Sun J, Perdew JP, Seidl M. Correlation energy of the 15:2819–2825. uniform electron gas from an interpolation between high- and low-density limits. Phys Rev B 2010, 62. Glasser ML. Pair correlation function for a two- 81:085123. dimensional electron gas. J Phys C: Solid State Phys 1977, 10:L121–L123. 80. Loos PF, Gill PMW. Correlation energy of the spin- polarized uniform electron gas at high density. Phys 63. Isihara A, Ioriatti L. Exact evaluation of the second- Rev B 2011, 84:033103. order exchange energy of a two-dimensional electron 81. Loos PF, Gill PMW. Leading-order behavior of the fluid. Phys Rev B 1980, 22:214–219. correlation energy in the uniform electron gas. Int J 64. Glasser ML. On some integrals arising in mathemati- Quantum Chem 2012, 112:1712–1716. – cal physics. J Comp App Math 1984, 10:293 299. 82. Loos P-F, Gill PMW. Thinking outside the box: the 65. Seidl M. Spin-resolved second-order correlation uniform electron gas on a hypersphere. J Chem Phys energy of the two-dimensional uniform electron gas. 2011, 135:214111. Phys Rev B 2004, 70:073101. 83. Loos PF. High-density correlation energy expansion 66. Chesi S, Giuliani GF. Correlation energy in a spin- of the one-dimensional uniform electron gas. J Chem polarized two-dimensional electron liquid in the high- Phys 2013, 138:064108. density limit. Phys Rev B 2007, 75:153306. 84. Wolfram Research, Inc.. Mathematica 7. 2008.

85. Loos PF. unpublished. 103. Ma F, Zhang S, Krakauer H. Finite-size correction in 86. Gill PMW, Loos PF, Agboola D. Basis functions for many-body electronic structure calculations of mag- electronic structure calculations on spheres. J Chem netic systems. Phys Rev B 2011, 84:155130. Phys 2014, 141:244102. 104. Ceperley DM. Fermion nodes. J Stat Phys 1991, – 87. Agboola D, Knol AL, Gill PMW, Loos PF. Uniform 63:1237 1267. electron gases. III. Low-density gases on three- 105. Bressanini D, Ceperley DM, Reynolds P. What do we dimensional spheres. J Chem Phys 2015, know about wave function nodes? In: Lester WA Jr, 143:084114. Rothstein SM, Tanaka S, eds. Recent Advances in 88. Wigner E. On the interaction of electrons in metals. Quantum Monte Carlo Methods, vol. 2. Singapore: fi – Phys Rev 1934, 46:1002–1011. World Scient c; 2001, 3 11. 106. Bajdich M, Mitas L, Drobny G, Wagner LK. Approx- 89. Loos PF, Gill PMW. Ground state of two electrons imate and exact nodes of fermionic wavefunctions: on a sphere. Phys Rev A 2009, 79:062517. coordinate transformations and topologies. Phys Rev 90. Aguilera-Navarro VC, Baker GA Jr, de Llano M. B 2005, 72:075131. Ground-state energy of jellium. Phys Rev B 1985, 107. Bressanini D, Reynolds PJ. Unexpected symmetry in 32:4502–4506. the nodal structure of the He atom. Phys Rev Lett 91. Fuchs K. A quantum mechanical investigation of the 2005, 95:110201. cohesive forces of metallic copper. Proc R Soc 1935, 108. Bressanini D, Morosi G, Tarasco S. An investigation 151:585–602. of nodal structures and the construction of trial wave 92. Carr WJ Jr. Energy, specific heat, and magnetic prop- functions. J Chem Phys 2005, 123:204109. erties of the low-density electron gas. Phys Rev 1961, 109. Mitas L. Structure of fermion nodes and nodal cells. 122:1437–1446. Phys Rev Lett 2006, 96:240402. 93. Kohn W, Schechter D. unpublished. 1961. 110. Scott TC, Luchow A, Bressanini D, Morgan JD III. 94. Carr WJ Jr, Coldwell-Horsfall RA, Fein AE. Anhar- Nodal surfaces of helium atom eigenfunctions. Phys monic contribution to the energy of a dilute electron Rev A 2007, 75:060101. gas: interpolation for the correlation energy. Phys 111. Bressanini D, Morosi G. On the nodal structure of – Rev 1961, 124:747 752. single-particle approximation based atomic wave 95. Bonsall L, Maradudin AA. Some static and dynamical functions. J Chem Phys 2008, 129:054103. properties of a two-dimensional Wigner crystal. Phys 112. Mitas L. Fermion nodes and nodal cells of noninter- – Rev B 1977, 15:1959 1973. acting and interacting fermions. arXiv:cond-mat:/ 96. Loos PF. Generalized local-density approximation 0605550. 2008. fi and one-dimensional nite uniform 34 electron gases. 113. Bressanini D. Implications of the two nodal domains Phys Rev A 2014, 89:052523. conjecture for ground state fermionic wave functions. 97. Foulkes WMC, Mitas L, Needs RJ, Rajagopal G. Phys Rev B 2012, 86:115120. Quantum Monte Carlo simulations of solids. Rev 114. Rasch KM, Mitas L. Impact of electron density on Mod Phys 2001, 73:33–83. the fixed-node errors in quantum Monte Carlo of 98. Kolorenc J, Mitas L. Applications of quantum Monte atomic systems. Chem Phys Lett 2012, 528:59–62. Carlo methods in condensed systems. Rep Prog Phys 115. Kulahlioglu AH, Rasch KM, Hu S, Mitas L. Density 2011, 74:026502. dependence of fixed-node errors in diffusion quantum 99. Fraser LM, Foulkes WMC, Rajagopal G, Needs RJ, Monte Carlo: triplet pair correlations. Chem Phys Kenny SD, Williamson AJ. Finite-size effects and cou- Lett 2014, 591:170–174. lomb interactions in quantum Monte Carlo calcula- 116. Rasch KM, Hu S, Mitas L. Fixed-node errors in tions for homogeneous systems with periodic quantum Monte Carlo: interplay of electron density boundary conditions. Phys Rev B 1996, and node nonlinearities. J Chem Phys 2014, 53:1814–1832. 140:041102. 100. Lin C, Zong FH, Ceperley DM. Twist-averaged 117. Loos PF, Bressanini D. Nodal surfaces and interdi- boundary conditions in continuum quantum Monte mensional degeneracies. J Chem Phys 2015, Carlo algorithms. Phys Rev E 2001, 64:016702. 142:214112. 101. Kwee H, Zhang S, Krakauer H. Finite-size correction 118. Ortiz G, Ballone P. Correlation energy, structure fac- in many-body electronic structure calculations. Phys tor, radial distribution function, and momentum dis- Rev Lett 2008, 100:126404. tribution of the spin-polarized uniform electron gas. – 102. Drummond ND, Needs RJ, Sorouri A, Phys Rev B 1994, 50:1391 1405. Foulkes WMC. Finite-size errors in continuum quan- 119. Ortiz G, Ballone P. Erratum: Correlation energy, tum Monte Carlo calculations. Phys Rev B 2008, structure factor, radial distribution function, and 78:125106. momentum distribution of the spin-polarized uniform

electron gas [Phys. Rev. B 50, 1391 (1994)]. Phys 136. Cioslowski J, Strasburger K, Matito E. The three- Rev B 1997, 56:9970. electron harmonium atom: the lowest-energy doublet 120. Ortiz G, Harris M, Ballone P. Zero temperature and quadruplet states. J Chem Phys 2012, phases of the electron gas. Phys Rev Lett 1999, 136:194112. 82:5317–5320. 137. Zhang S, Ceperley DM. Hartree-Fock ground state of 121. Kwon Y, Ceperley DM, Martin RM. Effects of back- the three-dimensional electron gas. Phys Rev Lett flow correlation in the three-dimensional electron gas: 2008, 100:236404. quantum Monte Carlo study. Phys Rev B 1998, 138. Bernu B, Delyon F, Duneau M, Holzmann M. Metal- 58:6800–6806. insulator transition in the 37 Hartree-Fock phase dia- 122. Zong FH, Lin C, Ceperley DM. Spin polarization of gram of the fully polarized homogeneous electron gas the low-density three-dimensional electron gas. Phys in two dimensions. Phys Rev B 2008, 78:245110. Rev E 2002, 66:036703. 139. Trail JR, Towler MD, Needs RJ. Unrestricted 123. Drummond ND, Towler MD, Needs RJ. Jastrow cor- Hartree-Fock theory of Wigner crystals. Phys Rev B relation factor for atoms, molecules, and solids. Phys 2003, 68:045107. Rev B 2004, 70:235119. 140. Bloch Z. Bemerkung zur elektronentheorie des ferro- 124. Spink GG, Needs RJ, Drummond ND. Quantum magnetismus und der elektrischen leitfähigkeit. Z Monte Carlo study of the three-dimensional spin- Physik 1929, 57:545–555. polarized homogeneous electron gas. Phys Rev B 141. Baguet L, Delyon F, Bernu B, Holzmann M. Hartree- 2013, 88:085121. Fock ground state phase diagram of jellium. Phys 125. Tanatar B, Ceperley DM. Ground state of the two- Rev Lett 2013, 111:166402. dimensional electron gas. Phys Rev B 1989, 142. Baguet L, Delyon F, Bernu B, Holzmann M. Proper- 39:5005–5016. ties of Hartree-Fock solutions of the three- 126. Kwon Y, Ceperley DM, Martin RM. Effects of three- dimensional electron gas. Phys Rev B 2014, body and backflow correlations in the two- 90:165131. dimensional electron gas. Phys Rev B 1993, 143. Delyon F, Duneau M, Bernu B, Holzmann M. Exist- – 48:12037 12046. ence of a metallic phase and upper bounds of the 127. Rapisarda F, Senatore G. Diffusion Monte Carlo Hartree-Fock energy in the homogeneous electron study of electrons in two-dimensional layers. Aust J gas. arXiv:0807.0770v1. 2008. – Phys 1996, 49:161 182. 144. Delyon F, Bernu B, Baguet L, Holzmann M. Upper 128. Attaccalite C, Moroni S, Gori-Giorgi P, Bachelet GB. bounds of spin-density wave energies in the homoge- Correlation energy and spin polarization in the 2D neous electron gas. Phys Rev B 2015, 92:235124. electron gas. Phys Rev Lett 2002, 88:256601. 145. Bernu B, Delyon F, Holzmann M, Baguet L. Hartree- 129. Attaccalite C, Moroni S, Gori-Giorgi P, Bachelet GB. Fock phase diagram of the two-dimensional electron Erratum: Correlation energy and spin polarization in gas. Phys Rev B 2011, 84:115115. the 2D electron gas [Phys. Rev. Lett. 88, 256601 146. Brown EW, Clark BK, DuBois JL, Ceperley DM. (2002)]. Phys Rev Lett 2003, 91:109902. Path-integral Monte Carlo simulation of the warm 130. Gori-Giorgi P, Attaccalite C, Moroni S, Bachelet GB. dense homogeneous electron gas. Phys Rev Lett Two-dimensional electron gas: correlation energy ver- 2013, 110:146405. sus density and spin polarization. Int J Quantum 147. Filinov VS, Fortov VE, Bonitz M, Moldabekov Z. – Chem 2003, 91:126 130. Fermionic path-integral Monte Carlo results for the 131. Drummond ND, Needs RJ. Phase diagram of the uniform electron gas at finite temperature. Phys Rev low-density two-dimensional homogeneous electron E 2015, 91:033108. gas. Phys Rev Lett 2009, 102:126402. 148. Schoof T, Groth S, Vorberger J, Bonitz M. Ab initio 132. Spivak B, Kivelson SA. Phases intermediate between a thermodynamic results for the degenerate electron gas two-dimensional electron liquid and Wigner crystal. at finite temperature. Phys Rev Lett 2015, Phys Rev B 2004, 70:155114. 115:130402. 133. Falakshahi H, Waintal X. Hybrid phase at the quan- 149. Brown EW, DuBois JL, Holzmann M, Ceperley DM. tum melting of the Wigner crystal. Phys Rev Lett Exchange-correlation energy for the three- 2005, 94:046801. dimensional homogeneous electron gas at arbitrary 134. Waintal X. On the quantum melting of the two- temperature. Phys Rev B 2013, 88:081102(R). dimensional Wigner crystal. Phys Rev B 2006, 150. Brown EW, DuBois JL, Holzmann M, Ceperley DM. 73:075417. Erratum: Exchange correlation energy for the three- 135. Clark BK, Casula M, Ceperley DM. Hexatic and dimensional homogeneous electron gas at arbitrary mesoscopic phases in a 2D quantum Coulomb sys- temperature [Phys. Rev. B 88, 081102(R) (2013)]. tem. Phys Rev Lett 2009, 103:055701. Phys Rev B 2013, 88:199901.